Linear Recurring Sequences Over Finite Fields

Citation

McEliece, Robert James (1967) Linear Recurring Sequences Over Finite Fields. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1KMK-T118. https://resolver.caltech.edu/CaltechETD:etd-10012002-154611

Abstract

This thesis deals with the problem of how the elements from a finite field F of characteristic p are distributed among the various linear recurrent sequences with a given fixed characteristic polynomial fε F[x]. The first main result is a method of extending the so-called "classical method" for solving linear recurrences in terms of the roots of f. The main difficulty is that f might have a root θ which occurs with multiplicity exceeding p-1; this is overcome by replacing the solutions θ^t, tθ^t, t²θ^t, ..., by the solutions θ^t, (t₁)θ^t, (t₂)θ^t, .... The other main result deals with the number N of times a given element a ε F appears in a period of the sequence, and for a≠0, the result is of the form N≡0 (mod p^ε where ε is an integer which depends upon f, but not upon the particular sequence in question. Several applications of the main results are given.

Thesis Files